arxiv: 2601.19597 · v4 · submitted 2026-01-27 · 💻 cs.LG

Recognition: 3 theorem links

· Lean Theorem

The Geometric Mechanics of Contrastive Representation Learning: Alignment Potentials, Entropic Dispersion, and Cross-modal Divergence

Yichao Cai , Zhen Zhang , Yuhang Liu , Javen Qinfeng Shi

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Pith reviewed 2026-05-16 11:09 UTC · model grok-4.3

classification 💻 cs.LG

keywords contrastive learningInfoNCEenergy landscapesgeometric bifurcationmeasure-theoretic frameworkmultimodal divergencealignment potentials

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The pith

In the large-batch limit contrastive learning converges to deterministic energy landscapes that bifurcate between unimodal and multimodal regimes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a measure-theoretic framework treating contrastive representation learning as the evolution of probability measures on a fixed embedding manifold. In the large-batch limit it proves that the stochastic InfoNCE objective converges in value and gradients to explicit deterministic energy functions. This convergence exposes a geometric bifurcation: unimodal regimes yield strictly convex energies with unique equilibria where entropy selects the solution, while multimodal regimes feature cross-coupled landscapes containing a negative symmetric divergence that allows aligned pairs to coexist with a persistent gap between modalities. Readers should care because this explains why models can achieve strong pairwise alignment yet retain modality-specific structures, shifting focus from pointwise losses to population-level geometry. Controlled experiments on synthetic data and CLIP representations confirm the predicted behaviors.

Core claim

In the large-batch limit the stochastic contrastive objective is shown to converge in both value and gradient to deterministic energy landscapes on the space of measures. These landscapes bifurcate into a unimodal case with strictly convex intrinsic energy admitting a unique Gibbs equilibrium and a symmetric multimodal case whose cross-coupled geometry includes a persistent negative symmetric divergence term, allowing strong pairwise alignment to persist alongside a modality gap.

What carries the argument

The deterministic energy landscapes obtained in the large-batch limit from evolving representation measures on a fixed embedding manifold under alignment potentials and entropic dispersion.

If this is right

Pairwise alignment is insufficient to control cross-modal marginal structure.
Entropy serves as a tie-breaker within the aligned basin in unimodal regimes.
Strong alignment can coexist with persistent modality gaps in multimodal settings.
The framework provides explicit geometric potentials that govern the training dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Designers of contrastive losses could target the cross-coupled divergence to reduce unwanted modality gaps.
The bifurcation may account for observed differences between image-text and single-modality contrastive tasks.
Testing the large-batch predictions on other architectures could reveal whether the manifold assumption holds in practice.

Load-bearing premise

Representation learning evolves measures on a fixed embedding manifold and the large-batch limit accurately captures the stochastic training dynamics without additional regularization effects.

What would settle it

A direct computation showing that the large-batch limit of the InfoNCE objective does not match the predicted deterministic energy landscape or that no bifurcation occurs in synthetic experiments with varying batch sizes.

Figures

Figures reproduced from arXiv: 2601.19597 by Javen Qinfeng Shi, Yichao Cai, Yuhang Liu, Zhen Zhang.

**Figure 1.** Figure 1: A unified analytical pipeline for contrastive learning geometry. Starting from stochastic InfoNCE losses, the large-batch limit yields deterministic parametric energies, which in turn lift to intrinsic free-energy landscapes over representation densities. At this intrinsic level, the geometry bifurcates: unimodally, the landscape is strictly convex, with entropy acting as a tie-breaker and yielding a Gibbs… view at source ↗

**Figure 2.** Figure 2: Numerical illustration of the geometric bifurcation. (a) In the unimodal intrinsic problem, the energy is Gibbs-like and concentrates toward low-potential regions as τ ↓. (b) In the multimodal setting, cross-modal coupling induces persistent marginal separation, whose magnitude grows with latent misalignment. intrinsic energy is strictly convex. Alignment dictates the ground-state basin, while entropy pro… view at source ↗

**Figure 3.** Figure 3: Large-batch gradient consistency across critics. Finitebatch InfoNCE gradients are compared against a high-fidelity large-batch reference as the number of negatives N increases. Left: gradient alignment with the reference (↑). Right: relative gradient error (↓). Results are shown for both the cosine critic in the spherical regime and the RBF critic in the compact Euclidean regime, with mean ± std over 20 … view at source ↗

**Figure 4.** Figure 4: Unimodal potential landscape on S 2 and equilibria across temperature. Left: the two-well potential U, with minima centers m1 and m2 marked. Right: Gibbs samples (blue) and trained particles (orange) for several temperatures τ . At high temperature, both distributions remain diffuse due to the stronger role of entropy; as τ decreases, both concentrate around the low-energy wells. −2 0 2 Angle a1 (modality … view at source ↗

**Figure 5.** Figure 5: Joint-angle coupling under controlled misalignment. Each panel shows a histogram of (a1, a2) for σmis ∈ {0.0, 0.1, 0.2, 0.3, 0.4}. When σmis = 0, the learned coupling is sharply concentrated near the diagonal, indicating a small modality gap. As σmis ↑, the diagonal band broadens and deforms, revealing a noisier cross-modal coupling while preserving the coarse latent modes. pare the resulting particle dist… view at source ↗

**Figure 6.** Figure 6: MS-COCO validation of the modality-gap mechanism. (a) Stronger average retrieval does not imply a smaller modality gap. Models with similar AvgR@1 can exhibit substantially different energy distance and centroid gap, indicating that retrieval quality and cross-modal geometric agreement are related but not equivalent. (b) As the same-category caption corruption probability p increases, average retrieval deg… view at source ↗

**Figure 7.** Figure 7: Unimodal potential landscape on S 2 and equilibria across temperature τ . Left top: the 2-well potential U in Eq. (74) (colored by value), with minima centers m1, m2. Others: Gibbs samples (blue; importance-resampled) and trained particles (orange; minimizing Eq. (76)) across various temperatures. As τ decreases, both distributions concentrate around the low-energy wells. with γ = 12, concentration κ = 12,… view at source ↗

**Figure 8.** Figure 8: Polar marginals across misalignment. Estimated angle marginals of the two modalities on S 1 . The mismatch becomes visually apparent for σmis > 0 and grows with σmis. the latent modes. As σmis increases, these clusters persist—indicating the model still captures the coarse latent semantics— but they deform into thickened, elongated ellipses. The diagonal band broadens significantly, reflecting that the con… view at source ↗

**Figure 9.** Figure 9: Joint-angle coupling across misalignment. 2D histograms of (a1, a2) from σmis = 0.0 to σmis = 0.7. Diagonal concentration (small σmis) indicates near-deterministic alignment; increasing off-diagonal spread (large σmis) indicates intrinsically noisy coupling [PITH_FULL_IMAGE:figures/full_fig_p051_9.png] view at source ↗

**Figure 10.** Figure 10: Residual embedding-space misalignment. Densities of ∆a = wrap(a2 −a1) from σmis = 0.0 to σmis = 0.7. The distribution broadens as σmis increases, indicating larger and more variable residual misalignment in representation space. 0.50 0.55 0.60 0.65 0.70 0.75 I→T Retrieval R@1 0.3 0.4 0.5 0.6 0.7 Energy gap Energy gap Centroid gap 0.60 0.65 0.70 0.75 0.80 0.85 0.90 Centroid gap RN50 ViT-B-16 RN101 RN50x16 … view at source ↗

**Figure 11.** Figure 11: Directional retrieval versus cross-modal gap on MS-COCO for pretrained OpenCLIP checkpoints. Left: image-to-text retrieval (I→T R@1) versus gap. Right: text-to-image retrieval (T→I R@1) versus gap. In both panels, the left y-axis shows energy distance and the right y-axis shows centroid gap. The relation is clearly non-monotone: models with similar directional retrieval can exhibit substantially different… view at source ↗

**Figure 12.** Figure 12: Directional retrieval under same-category caption corruption on MS-COCO. Left: image-to-text retrieval (I→T R@1) versus corruption probability p. Right: text-to-image retrieval (T→I R@1) versus p. Bars show the centroid gap on the right axis. For both RN50 and ViT-B-16, increasing same-category mispairing degrades retrieval in both directions while systematically enlarging the modality gap. F.4.2. CONTROL… view at source ↗

read the original abstract

While InfoNCE underlies modern contrastive learning, its geometric mechanisms remain under-characterized beyond the canonical alignment--uniformity decomposition. We develop a measure-theoretic framework in which learning evolves representation measures on a fixed embedding manifold. In the large-batch limit, we prove value and gradient consistency, linking the stochastic objective to explicit deterministic energy landscapes and revealing a geometric bifurcation between unimodal and symmetric multimodal regimes. In the unimodal case, the intrinsic energy is strictly convex and admits a unique Gibbs equilibrium, showing that entropy acts as a tie-breaker within the aligned basin. In the multimodal case, the intrinsic geometry becomes cross-coupled and contains a persistent negative symmetric divergence term: each modality's marginal reshapes the effective landscape of the other, allowing strong pairwise alignment to coexist with a persistent modality gap. Controlled synthetic experiments and analyses of pretrained CLIP representations support these predictions. Overall, our results shift the analytical lens from pointwise discrimination to population geometry, showing that pairwise alignment alone is insufficient to control cross-modal marginal structure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a measure-theoretic account of contrastive learning that derives explicit energy landscapes in the large-batch limit and identifies a geometric split between unimodal convex equilibria and multimodal regimes with a persistent negative symmetric divergence term.

read the letter

The main takeaway is that this work formalizes contrastive objectives as evolving measures on a fixed embedding manifold and shows, in the large-batch limit, that the stochastic InfoNCE objective converges in value and gradient to deterministic energy landscapes. That limit produces a clean bifurcation: unimodal cases yield strictly convex intrinsic energies with a unique Gibbs equilibrium where entropy breaks ties inside the aligned basin, while multimodal cases introduce cross-coupling and a negative symmetric divergence term that lets strong pairwise alignment coexist with a lasting modality gap. The synthetic experiments and the checks against pretrained CLIP representations give direct support for these predictions and move the analysis past the standard alignment-uniformity split to population geometry. That shift is the clearest new element. The framework is built from energy functionals and measure theory without obvious circular fitting, and the large-batch consistency claim is stated as an external approximation rather than an internal reduction. The experiments are controlled enough to test the bifurcation idea on both synthetic data and real models. The soft spot is the large-batch limit itself. Finite-batch SGD, momentum, and weight decay introduce implicit regularization that can change marginal dispersion and cross-modal terms, so the deterministic landscapes may not govern actual trajectories without an error bound that accounts for those effects. The fixed-manifold assumption also leaves open how much the geometry changes during training. I would want to see the explicit derivations for convexity and consistency before treating the bifurcation as robust. This paper is for researchers working on the theoretical side of contrastive and multimodal representation learning who already know the alignment-uniformity literature and want a geometric extension. A reader looking for new analytical tools to think about vision-language models would find the bifurcation and the negative divergence term useful to discuss. It deserves a serious referee because the claims are specific, the experiments are present to check them, and the framework is coherent on its own terms even if the approximation step needs tightening.

Referee Report

1 major / 2 minor

Summary. The manuscript develops a measure-theoretic framework for contrastive representation learning in which representations evolve as measures on a fixed embedding manifold. In the large-batch limit the authors prove value and gradient consistency between the stochastic InfoNCE objective and explicit deterministic energy landscapes, revealing a geometric bifurcation: unimodal regimes yield strictly convex intrinsic energies with unique Gibbs equilibria (entropy acting as tie-breaker), while multimodal regimes exhibit cross-coupled geometry containing a persistent negative symmetric divergence term that permits strong pairwise alignment alongside modality gaps. The predictions are supported by controlled synthetic experiments and analyses of pretrained CLIP representations.

Significance. If the consistency proofs and bifurcation analysis hold, the work supplies a useful geometric lens that moves beyond the alignment-uniformity decomposition to population-level marginal structure. The explicit energy functionals and the distinction between unimodal and cross-coupled multimodal regimes provide explanatory power for observed modality gaps and falsifiable predictions about regime transitions. The combination of measure-theoretic derivations with targeted experiments is a strength.

major comments (1)

[§3] §3 (large-batch limit and consistency proofs): The value and gradient consistency between the stochastic objective and the deterministic energy landscapes is derived under the assumption that the limit introduces no extra regularization. Finite-batch SGD, momentum, and weight decay induce implicit regularization that can reshape marginal dispersion and the cross-modal divergence term, potentially shifting or removing the predicted bifurcation. An explicit error bound that accounts for these effects is required to confirm that the deterministic landscapes govern actual training trajectories.

minor comments (2)

[§2] Notation for the symmetric divergence term and the intrinsic energy functional should be introduced with a single consolidated definition rather than scattered across the text.
[§5] The synthetic experiment section would benefit from an explicit statement of how the embedding manifold is held fixed during optimization and how the empirical measures are constructed to match the theoretical setup.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address the major comment regarding the large-batch limit below.

read point-by-point responses

Referee: [§3] §3 (large-batch limit and consistency proofs): The value and gradient consistency between the stochastic objective and the deterministic energy landscapes is derived under the assumption that the limit introduces no extra regularization. Finite-batch SGD, momentum, and weight decay induce implicit regularization that can reshape marginal dispersion and the cross-modal divergence term, potentially shifting or removing the predicted bifurcation. An explicit error bound that accounts for these effects is required to confirm that the deterministic landscapes govern actual training trajectories.

Authors: We appreciate the referee highlighting this important distinction. Our value and gradient consistency results in §3 are derived strictly in the large-batch limit, where the empirical measure converges to the population measure and the stochastic InfoNCE objective converges to the deterministic energy landscape without sampling-induced regularization. We agree that finite-batch SGD, momentum, and weight decay introduce implicit regularization capable of reshaping marginal dispersion and the cross-modal divergence term. Our controlled synthetic experiments in Section 5 and the CLIP analyses were performed under standard finite-batch training with momentum and weight decay; these experiments show that the predicted unimodal convexity and multimodal negative symmetric divergence persist qualitatively. We will add a dedicated discussion paragraph clarifying the scope of the large-batch analysis, acknowledging the role of implicit regularization, and noting that quantitative error bounds between finite-batch trajectories and the deterministic limit are left for future work. This constitutes a partial revision. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external measure-theoretic limits

full rationale

The paper constructs a measure-theoretic framework for representation measures on a fixed embedding manifold and derives value/gradient consistency between the stochastic InfoNCE objective and deterministic energy landscapes strictly in the large-batch limit. This consistency is obtained via mathematical limit arguments rather than by redefining any quantity in terms of itself or by fitting parameters inside the target equations. The subsequent geometric bifurcation between unimodal and multimodal regimes follows directly from the convexity properties and cross-coupling terms of the derived functionals, without importing uniqueness from prior self-citations or smuggling ansatzes. No load-bearing step reduces to a fitted input renamed as prediction or to a self-referential definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on a measure-theoretic setup and large-batch consistency that are introduced without additional fitted constants beyond standard contrastive hyperparameters.

axioms (2)

domain assumption Representation learning evolves measures on a fixed embedding manifold
Invoked when defining the intrinsic energy landscapes.
domain assumption Large-batch limit yields value and gradient consistency between stochastic and deterministic objectives
Central link used to derive the energy landscapes and bifurcation.

invented entities (2)

Intrinsic energy landscape no independent evidence
purpose: Deterministic functional whose minima govern large-batch contrastive dynamics
Introduced to replace the stochastic objective in the limit analysis.
Symmetric divergence term no independent evidence
purpose: Cross-modal interaction that produces persistent modality gap in multimodal regime
New term appearing in the multimodal energy expression.

pith-pipeline@v0.9.0 · 5490 in / 1433 out tokens · 51789 ms · 2026-05-16T11:09:11.110368+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the intrinsic unimodal energy Fτ,U(ρ) := (1/τ)∫U(z)ρ(z)dμ(z) − H(ρ) is strictly convex … admits a unique minimizer ρ∗(z) = exp(−U(z)/τ)/Zτ (Prop. 3.1)
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

large-batch value and gradient consistency … linking the stochastic objective to explicit deterministic energy landscapes (Thms. 3.1 and 4.1)
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

negative symmetric divergence coupling … each modality’s marginal acts as a logarithmic barrier

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Works this paper leans on

14 extracted references · 14 canonical work pages · 3 internal anchors

[1]

Chen, T., Luo, C., and Li, L

https://proceedings.mlr.press/ v119/chen20j/chen20j.pdf. Chen, T., Luo, C., and Li, L. Intriguing properties of con- trastive losses.Advances in Neural Information Process- ing Systems, 34:11834–11845, 2021. https://dl. acm.org/doi/10.5555/3540261.3541166. Chuang, C.-Y ., Robinson, J., Yen-Chen, L., Torralba, A., and Jegelka, S. Debiased contrastive learn...

work page doi:10.5555/3540261.3541166 2021
[2]

2017.12.012

https://doi.org/10.1016/j.neunet. 2017.12.012. Gutmann, M. U. and Hyv ¨arinen, A. Noise-contrastive estimation: A new estimation principle for unnormal- ized statistical models. In Teh, Y . W. and Tittering- ton, M. (eds.),Proceedings of the 13th International Conference on Artificial Intelligence and Statistics (AIS- TATS 2010), volume 9 ofProceedings of...

work page doi:10.1016/j.neunet 2017
[3]

Gaussian Error Linear Units (GELUs)

https://openreview.net/forum?id= AuEgNlEAmed. Hendrycks, D. Gaussian error linear units (gelus).arXiv preprint arXiv:1606.08415, 2016. https://arxiv. org/pdf/1606.08415. Hyvarinen, A. and Morioka, H. Unsupervised feature extraction by time-contrastive learning and nonlinear ica.Advances in neural information processing sys- tems, 29, 2016. https://dl.acm....

work page internal anchor Pith review Pith/arXiv arXiv doi:10.5555/3157382.3157519 2016
[4]

Adam: A Method for Stochastic Optimization

PMLR, 2021. https://proceedings.mlr. press/v139/jia21b/jia21b.pdf. Kingma, D. P. and Ba, J. Adam: A method for stochastic optimization. In3rd International Conference on Learn- ing Representations (ICLR), 2015. https://arxiv. org/pdf/1412.6980. Lei, Y . and Ying, Y . Fine-grained analysis of stability and generalization for stochastic gradient descent. In...

work page internal anchor Pith review Pith/arXiv arXiv 2021
[5]

Representation Learning with Contrastive Predictive Coding

PMLR, 2020. https://proceedings.mlr. press/v119/lei20c/lei20c.pdf. Liang, V . W., Zhang, Y ., Kwon, Y ., Yeung, S., and Zou, J. Y . Mind the gap: Understanding the modality gap in multi-modal contrastive representation learning.Ad- vances in Neural Information Processing Systems, 35: 17612–17625, 2022. https://openreview.net/ forum?id=S7Evzt9uit3. Liao, H...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/978-3-319-10602-1_48 2020
[6]

https://proceedings.mlr

PMLR, 2019. https://proceedings.mlr. press/v97/poole19a.html. Radford, A., Kim, J. W., Hallacy, C., Ramesh, A., Goh, G., Agarwal, S., Sastry, G., Askell, A., Mishkin, P., Clark, J., et al. Learning transferable visual models from natural language supervision. InInter- national conference on machine learning, pp. 8748–

work page 2019
[7]

https://proceedings.mlr

PmLR, 2021. https://proceedings.mlr. press/v139/radford21a/radford21a.pdf. Ryu, J. J., Yeddanapudi, P., Xu, X., and Wornell, G. W. Contrastive predictive coding done right for mutual in- formation estimation. InThe Fourteenth International Conference on Learning Representations, 2026. https: //openreview.net/forum?id=JodkBXWgbA. Saunshi, N., Ash, J., Goel...

work page 2021
[8]

Shi, P., Welle, M

https://proceedings.mlr.press/ v162/saunshi22a/saunshi22a.pdf. Shi, P., Welle, M. C., Bj¨orkman, M., and Kragic, D. Towards understanding the modality gap in clip. InICLR 2023 workshop on multimodal representation learning: perks and pitfalls, 2023. https://openreview.net/ pdf?id=8W3KGzw7fNI. Sun, J., Zhang, S., Li, H., and Wang, M. Contrastive learn- ing...

work page arXiv 2023
[9]

Uesaka, T., Suzuki, T., Takida, Y ., Lai, C.-H., Murata, N., and Mitsufuji, Y

https://openreview.net/forum?id= rkxoh24FPH. Uesaka, T., Suzuki, T., Takida, Y ., Lai, C.-H., Murata, N., and Mitsufuji, Y . Weighted point set embedding for multimodal contrastive learning toward optimal sim- ilarity metric. InThe Thirteenth International Con- ference on Learning Representations, 2025. https: //openreview.net/forum?id=uSz2K30RRd. V on K¨...

work page doi:10.5555/3524938.3525859 2025
[10]

Yao, D., Xu, D., Lachapelle, S., Magliacane, S., Taslakian, P., Martius, G., von K¨ugelgen, J., and Locatello, F

https://www.stats.ox.ac.uk/˜teh/ research/compstats/WelTeh2011a.pdf. Yao, D., Xu, D., Lachapelle, S., Magliacane, S., Taslakian, P., Martius, G., von K¨ugelgen, J., and Locatello, F. Multi- view causal representation learning with partial observ- ability. InThe Twelfth International Conference on Learn- ing Representations, 2024. https://openreview. net/f...

work page 2024
[11]

Yoshida, N., Hayakawa, S., Takida, Y ., Uesaka, T., Wakaki, H., and Mitsufuji, Y

https://openreview.net/forum?id= UUAjF4xL0e. Yoshida, N., Hayakawa, S., Takida, Y ., Uesaka, T., Wakaki, H., and Mitsufuji, Y . Theoretical refinement of clip by utilizing linear structure of optimal similarity.arXiv preprint arXiv:2510.15508, 2025. https://arxiv. org/pdf/2510.15508. Zhai, X., Mustafa, B., Kolesnikov, A., and Beyer, L. Sigmoid loss for la...

work page arXiv 2025
[12]

Zhang, G., Wang, C., Xu, B., and Grosse, R

https://openaccess.thecvf.com/ content/ICCV2023/papers/Zhai_Sigmoid_ Loss_for_Language_Image_Pre-Training_ ICCV_2023_paper.pdf. Zhang, G., Wang, C., Xu, B., and Grosse, R. Three mech- anisms of weight decay regularization. InInternational Conference on Learning Representations, 2019. https: //openreview.net/forum?id=B1lz-3Rct7. Zimmermann, R. S., Sharma, ...

work page 2019
[13]

content” from view-varying “style

https://proceedings.mlr.press/ v139/zimmermann21a/zimmermann21a.pdf. 12 The Geometric Mechanics of Contrastive Learning Appendix Contents A. Related Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

work page 2010
[14]

Empirically, this manifests as a geometric tug-of-war between the modalities, yielding two distinct observable effects: (i) The broadening diagonal band in Fig

Consequently, the two directions of the symmetric objective cannot be governed by a single, unified potential function. Empirically, this manifests as a geometric tug-of-war between the modalities, yielding two distinct observable effects: (i) The broadening diagonal band in Fig. 9 reflects the necessary rise in conditional entropy driven by the mismatche...

work page 2021